Jian-Wei Pan Decoherence-free sub-space and quantum error-rejection Jian-Wei Pan Lecture Note 7

Jian-Wei PanDecoherence open system dynamics System Environment The off-diagonal element of the qubit density matrix will drop down with the rate depends on the coupling between qubit and environment. More generally... How to guide the dynamics of system-environment coupling?

Jian-Wei Pan  Quantum Error Correction for QC  Active (Error correction): deal well with independent errors on qubits  Quantum Entanglement Purification for QC  Entanglement Purification (any unknown mixed state)  Local Filtering (known state)  Entanglement Concentration (unknown state)  QC based on Decoherence-free Subspace  Passive (error avoidance): find a subspace of the system space over which evolution stays unitary, unperturbed, correlated noise  Error-free Transfer in QC  Active (error rejection): reject the contaminated information Possible solutions to overcome decoherence in long-distance quantum communication (QC)

Jian-Wei Pan  QC based on Decoherence- free Subspace  Error-free Transfer in QC

Jian-Wei Pan Decoherence-free subspace (DFS)

Jian-Wei Pan Decoherence Free Subspace General Definitions, Collective Decoherence Robustness to perturbing error processes Use of DF subspace for concatenation into a Quantum Error Correcting Code (QECC) Relationship between DF subspace and QECC Existential universality results on DF subspaces/symmetrization methods Subsystem Generalization How do we perform quantum communication in a DFS? [Phys. Rev. Lett. 79, 1953 (1997); Phys. Rev. Lett. 79, 3306 (1997); Phys. Rev. Lett. 81, 2594 (1998)] [Phys. Rev. Lett. 81, 2594 (1998); Phys. Rev. A 60, 1944 (1999)] [Phys. Rev. Lett. 82, 4556 (1999)] [Phys. Rev. A (R) (1999)] [Phys. Rev. Lett. 84, 2525(2000)] 1997 Symmetrization/Bang-bang methods [Phys. Rev. A 58, 2733 (1998); Phys.Lett. A 258, 77 (1999) ] DFS History

Jian-Wei Pan DFS under Collective Noise Collective Rotation Noise ： Noise can be seen as some unitary transformation as U(θ,Φ), if for all the channel, the unitary is the same, then it is called collective noise. If Φ is 0, i.e., U ＝ U(θ), it is called collective rotation noise

Jian-Wei Pan [P. G. Kwiat et al., Science 290, 498(2000); J. B. Altepeter, et al., Phys. Rev. Lett. 92, (2004)] DFS under Collective Rotation Noise

Jian-Wei Pan DFS for Collective Rotation Noise The two state are invariant under the collective rotation noise. All the linear superposition of the two states constitute a subspace that is decoherence free to the noise. [P. G. Kwiat et al., Science 290, 498(2000);

Jian-Wei Pan Similar to BB84, +,- respect the diagonal state and anti-diagonal state respectively. The four state can be used to encode key and the security bound is the same as BB84 protocol. Application in quantum key distribution using a DFS [X.B.Wang, Phys. Rev. A 72, (R) (2005)]

Jian-Wei Pan Experimental Setup [Q. Zhang, PRA 73, (R) 2006]

Jian-Wei Pan Experimental Result QBER of DFS and traditional BB84 under the collective rotation noise. |θ| > π/8, QBER BB84 >11%

Jian-Wei PanDrawback DFS only for Collective Rotation Noise Other noise  Free space phase drifting caused by temperature difference  Long distance in optical fibers will cause a redoubtable obstacle Noise not only in H/V basis!

Jian-Wei Pan Collective Noise

Jian-Wei Pan A new protocol First apply a time delay between H and V, the state will be After a collective noise Bob can measure in any direction (H’/V’) which also can be considered as part of the collective noise.

Jian-Wei Pan A new protocol Then again, Bob apply a time delay between H and V, the state will be The last operation is to project the state onto the subspace in which the photons arrive exactly at the same time

Jian-Wei Pan We will get with a probability  1/3 by a random unitary transformation A new protocol

Jian-Wei Pan [T.-Y Chen et al., Phys. Rev. Lett (2006)] Experimental Setup

Jian-Wei Pan Experimental Result 4m fiber without random rotations with random rotations average QBER

Jian-Wei Pan Experimental Result 1km fiber [T.-Y Chen et al., Phys. Rev. Lett (2006)] without random rotations with random rotations average QBER

Jian-Wei Pan  QC based on Decoherence- free Subspace  Error-free Transfer in QC

Jian-Wei Pan

Bit-flip Error Correction CNot Operation Required!!! [D. Bouwmeester, PRA 63, (R) (2001).] two bits flipping (p 2 ) can’t be corrected

Jian-Wei Pan Error-free transfer

Jian-Wei Pan Problem in Experimental Realization Possibility of two pair emission is in the same order and will cause four-fold coincidence!

Jian-Wei Pan [X.-B. Wang, PRA 69, (2004)] Error-free transfer 2’ 1’2” 1”

Jian-Wei Pan Through a noisy channel with bit-flip error rate p new the remaining QBER will be

Jian-Wei Pan Experimental Set-up Trigged by D 4 possibility of two pair emission will be much lower [Y.-A. Chen et al., PRL 96, (2006)]

Jian-Wei Pan By one HWP inside two QWP, any U-transmit can be implemented! Bit-flip-error simulation

Jian-Wei Pan

Quantum Noisy Channel

Jian-Wei Pan Experimental Results [Y.-A. Chen et al., PRL 96, (2006)]

Jian-Wei Pan The phase-shift error rejection can be realized.

Jian-Wei Pan The higher order bit-flip error can be rejected. encoding unknown quantum states into higher multi- photon entanglement (N), the higher order (up to N-1) error can be rejected

Jian-Wei Pan Applied to the quantum key distribution the threshold of tolerable error rate over the quantum noisy channel can be greatly improved. [X.-B. Wang, PRL 92, (2004)]